Skip to main content

Gaussian Integral

The Gaussian integral is the integral of the function ex2e^{-x^2} over the entire real line, and it is given by:

ex2dx=π\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}

First, we square the integral:

(ex2dx)2=ex2ey2dxdy\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2} e^{-y^2} dx dy

Next, we change to polar coordinates:

002πer2rdθdr=π\int_{0}^{\infty} \int_{0}^{2\pi} e^{-r^2} r d\theta dr = \pi

To evaluate this integral, we use the substitution u=r2u = r^2 and du=2rdrdu = 2r dr, which gives:

120eudu=12\frac{1}{2}\int_{0}^{\infty} e^{-u} du = \frac{1}{2}

Therefore, we have:

(ex2dx)2=πex2dx=π\left(\int_{-\infty}^{\infty} e^{-x^2} dx\right)^2 = \pi \quad \Rightarrow \quad \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}

And this completes the proof.